Nonlinear analysis of flow in an elastic tube (artery): steady streaming effects

We analyse the nonlinear flow of a Newtonian fluid in an elastic tube when subjected to an oscillatory pressure gradient with motivation from the problem of blood flow in arteries. Two parameters: the unsteadiness, α = R 0 (ω/ν) ½ and the diameter variation, ε = ( R max − R 0 )/ R 0 , are important in characterizing the flow problem. The diameter variation (ε) is taken to be small so that the perturbation method is valid, and asymptotic solutions for two limiting cases of the steady-streaming Reynolds number, R s = (αε) 2 (either small or large), are derived. The results indicate that nonlinear convective acceleration induces finite mean pressure gradient and mean wall shear rate even when no mean flow occurs. The magnitude of this effect depends on the amplitude of the diameter variation and the flow rate waveforms and the phase angle difference between them, which can be related to the impedance (pressure/flow) phase angle. Changes in the impedance phase angle, which is indicative of the degree of wave reflection, can change the direction of the induced mean flow. It is also shown that the induced mean wall shear rate is proportional to α when α is large. In addition, it is observed that the steady flow structure in the core can be influenced by wave reflection. The streamlines in the core are always parallel to the tube wall when there is no reflection. However, with total reflection, the induced mean flow recirculates between the nodes and points of maximum amplitude in a closed streamline pattern. Implications of the steady-streaming phenomena for physiological flow applications are discussed in a concluding section.